6 3 points If there are no restrictions on where the digits

6. (3 points) If there are no restrictions on where the digits and letters are placed, how many 8-place license plates consisting of 5 letters and 3 digits are possible if no repetitions of letters or digits are allowed and the 3 digits must be consecutive?

Solution

8-place license plates consisting of 5 letters and 3 digits are possible if no repetitions of letters or digits must be consecutive are possible in below case - We have 10 number of digits , 26 alphabets are avalible
---AAAAA, 3 consecutive is filled with 10 C 3 * 26 C 5
A---AAAA, 3 consecutive is filled with 10 C 3 * 26 C 5
AA---AAA, 3 consecutive is filled with 10 C 3 * 26 C 5
AAA---AA, 3 consecutive is filled with 10 C 3 * 26 C 5
AAAA---A, 3 consecutive is filled with 10 C 3 * 26 C 5
AAAAA---, 3 consecutive is filled with 10 C 3 * 26 C 5

The total No.of such possibles are = 6 * 10 C 3 * 26 C 5 = = nCr(10,3)*nCr(26,5)*6 = 47361600